Solution of Gamma function of 2n
Answers
The (complete) gamma function Gamma(n) is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by
Gamma(n)=(n-1)!,
(1)
a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler Pi(n)=n! (Gauss 1812; Edwards 2001, p. 8).
It is analytic everywhere except at z=0, -1, -2, ..., and the residue at z=-k is
Res_(z=-k)Gamma(z)=((-1)^k)/(k!).
(2)
There are no points z at which Gamma(z)=0.
The gamma function is implemented in the Wolfram Language as Gamma[z].
There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use Gamma^n(z) (i.e., using a trigonometric function-like convention), it is also common to write [Gamma(z)]^n.
The gamma function can be defined as a definite integral for R[z]>0 (Euler's integral form)
Gamma(z) = int_0^inftyt^(z-1)e^(-t)dt
(3)
= 2int_0^inftye^(-t^2)t^(2z-1)dt,
(4)
or
Gamma(z)=int_0^1[ln(1/t)]^(z-1)dt.
(5)
The complete gamma function Gamma(x) can be generalized to the upper incomplete gamma function Gamma(a,x) and lower incomplete gamma function gamma(a,x).
GammaReImAbs
Min Max
Re
-5
5
Im
-5
5
Plots of the real and imaginary parts of Gamma(z) in the complex plane are illustrated above.